A certain sum becomes Rs. 2286 in 3 years and Rs. 2448 in 4 years at simple interest. Using these two consecutive amounts under simple interest, what is the annual rate of interest in percentage?

Introduction / Context:
In this simple interest question, we know the amount after 3 years and after 4 years. Because simple interest adds the same interest every year, the difference between the amounts for years 3 and 4 tells us the interest for one year. We can then use that yearly interest to find the principal and compute the annual interest rate as a percentage.

Given Data / Assumptions:

• Amount after 3 years A3 = Rs. 2286.
• Amount after 4 years A4 = Rs. 2448.
• Principal P remains constant.
• Simple interest applies, so yearly interest is constant.
• Simple interest formula: A = P + (P * R * T) / 100.

Concept / Approach:
The difference A4 - A3 represents the interest for one extra year. Once we know the yearly interest, we compute the total 3-year interest and subtract it from A3 to get the principal P. Finally, we use SI for one year and the simple interest formula to solve for the annual rate R.

Step-by-Step Solution:
Difference between A4 and A3: 2448 - 2286 = Rs. 162. This Rs. 162 is the simple interest for 1 year. Thus, yearly interest I_year = Rs. 162. Total interest for 3 years = 3 * 162 = Rs. 486. Principal P = A3 - 3-year interest = 2286 - 486 = Rs. 1800. Now, for 1 year: SI = 162 = (P * R * 1) / 100 = 1800 * R / 100. So 162 = 18R, giving R = 162 / 18 = 9% per annum.
Verification / Alternative check:
Check with P = 1800 and R = 9%. Amount after 3 years: A3 = 1800 * (1 + 9 * 3 / 100) = 1800 * (1 + 0.27) = 1800 * 1.27 = 2286, matching the given A3. Amount after 4 years: A4 = 1800 * (1 + 9 * 4 / 100) = 1800 * 1.36 = 2448, which matches A4. This confirms that R = 9%.

Why Other Options Are Wrong:
Rates of 8%, 10%, 11%, or 12% produce yearly interest values different from Rs. 162 on the principal that fits A3 and A4. Substituting those rates into the formula would lead to inconsistent amounts that do not match both given values for 3 and 4 years simultaneously.

Common Pitfalls:
Some students mistakenly divide the total difference between 4-year and 3-year amounts by the entire 4 years, rather than recognizing that it represents just one year of interest. Others try to find R directly without isolating the principal. Using the stepwise method of finding annual interest, then principal, then rate, helps avoid errors.

Final Answer:
The annual rate of interest is 9% per annum.